This study explores the development of early algebraic reasoning in mathematical communities of inquiry. Under consideration is the different pathways teachers take as they develop their own understanding of early algebra and then enact changes in their classroom to facilitate algebraic reasoning opportunities. Teachers participated in a professional development intervention which focused on understanding of early algebraic concepts, task development, modification, and enactment, and classroom and mathematical practices. Design research was employed to investigate both teaching and learning in the naturalistic setting of the schools and classrooms. The design approach supported the development of a model of professional development and the framework of teacher actions to facilitate algebraic reasoning. Data collection over the school year included participant observations, video recorded observations, documents, teacher interviews, and photo elicitation interviews with students. Retrospective data analysis drew the results together to be presented as cases of two teachers, their classrooms, and students. The findings show that the integration of algebraic reasoning into classroom mathematical activity is a gradual process. It requires teachers to develop their own understanding of algebraic concepts which includes understanding of student reasoning, progression, and potential misconceptions. Task implementation and design, shifts in pedagogical actions, and the facilitation of new classroom and mathematical practices were also key elements of change. The important role which students have in the development of classrooms where algebraic reasoning is a focus was also highlighted. These findings have significant implications for how teachers can be supported to develop their understanding of early algebra and use this understanding in their own classrooms to facilitate early algebraic reasoning.

This thesis reports empirical and theoretical research into learning of mathematics and statistics at university level, with particular regard to students views of their self-confidence and experiences, and the effects of these on achievement. This study was conducted at a time of widespread national concern about difficulties in mathematics education in England, particularly at the transition from school to university Science, Technology, Engineering and Mathematics (STEM) courses. Factors which affected non-specialist students learning of mathematics and statistics were investigated using student surveys in 2004/5, 2005/6 and 2006/7 (701 questionnaires) in the a-typical setting of a University College specialising in rural and land-based higher education. 52 student interviews were also carried out, primarily in 2008 and 2009, and are referred to but are not the main focus of this thesis. Both deductive and inductive approaches were used. Self-confidence was defined using three Mathematics Self-confidence Domains: Overall Confidence in Mathematics, Topic confidences for specific tasks, and Applications Confidence. Self-confidence was considered a belief, whilst liking of the subjects was an attitude, both forming part of affect , where affect comprised beliefs, attitudes and emotions. Student motivation was also investigated. The survey data, and examination and assignment marks, of engineering students learning mathematics and other non-specialist students learning statistics, were analysed both quantitatively (by descriptive statistics, ANOVA, Kruskal Wallis, Correlation, Multiple Regression, Factor and Cluster analyses) and qualitatively. Previous success in mathematics, primarily GCSE Mathematics grade, was found to be the greatest determinant of university students success in mathematics and statistics, but self-confidence and other affective variables also had significantly measurable effects. Significant effects on student confidence were also found for gender and dyslexia despite good achievement. Findings indicate that students self-confidence in mathematics does matter, as evidenced by significant relationships between confidence and achievement, but it was also concluded that these inter-relations were complex. Educators are encouraged to adopt student-focussed teaching styles which improve students self-confidence as a means to improving attainment.

Girls' relationship with mathematics has been an extensive and contested area of investigation during the last 40 years, mainly in developed countries. This contrasts with the small amount of research from developing countries, where the topic has been largely neglected but may present different challenges. In Chile, such lack of empirical evidence is surprising, particularly because of several national reports describing attainment differences in the national assessment test (SIMCE), where girls are consistently outperformed by boys. Currently, there are no studies which systematically explore gender differences in attainment in Chile. In addition, only a small number of studies have tried to explain why these differences, as well as others in engagement, attitudes and enrolment in mathematics, arise in this country. The main goal of this thesis is to critically examine these issues by investigating how girls relate to mathematics during early adolescence in Chile, and how such relationships are influenced/mediated by certain social variables (e.g. social class, classroom cultures and peer group identities).In order to do this, this thesis has adopted a mixed methods approach, thus linking analysis and results from studies that use both quantitative and qualitative methodologies. Firstly, I investigate the size and distribution of the gender attainment gap in Mathematics in Chile using a Multilevel approach to analyse data from the national census of educational quality (SIMCE). Here, I analyse the naturalization of gender differences based on results, and conclude that differences found in attainment between boys and girls are small and dependent on socioeconomic status. I then explore how girls' subjective relationships with mathematics are constructed, and how different social influences mediate this process. Using the concept of Mathematical Identities [MIs] as a main tool I explore the influence of social variables on the construction of girls' MIs in Chilean classrooms and I also consider how teaching practices and peer social relations in the classroom mediate these identities. A key finding here is the positive relationship between students' perceptions of their teaching as student-centred and more positive MI, which is in fact the same for girls and boys. A second key finding is that both representational and enacted aspects of girls' MI are mediated by their relationship with peers and peer groups. This mediation process can be described as a negotiation of different forms of belonging to social groups, which involved also the negotiation of different MIs inside the classroom. The main conclusion of this thesis is that in order to understand the role of gender in mediating girls' relationships with mathematics, we need to acknowledge the profoundly situated nature of this relationship in the cultural practices of the classroom, including mathematical practices, but also peer group practices. This argues against discourses that essentialise and naturalize 'gendered relationships with mathematics' which appear to be pre-dominant in the collation of national assessment data (like SIMCE) where categories such as gender, class, ethnicity etc. are viewed as causal or explanatory rather than produced 'in practice'.

Recent reports on mathematics education in English secondary schools have consistently expressed concern about students' performance and enjoyment as well as their progression into studying mathematics post-16 (Smith, 2004; Ofsted, 2006, 2008a; Royal Society, 2008, 2010; Vorderman et al, 2011). Too often students were expected to follow rules and procedures without mastering underlying concepts and connections, and hence without developing their mathematical understanding (Ofsted 2008a). Boaler (2008a) provides evidence for the introduction of Complex Instruction (CI) as an effective alternative approach to teaching and learning mathematics. The CI pedagogy combines rich mathematical tasks and instructional strategies that foster collaborative group work and problem solving. The approach emphasises effort over ‘ability' and challenges beliefs that only some students can do mathematics and that they should be taught in ‘ability' groups. This thesis explores factors which facilitate or militate against the adoption of such an approach by drawing upon Stigler and Hiebert's (1999) concept of a ‘cultural script' and Dweck's (2000) ‘theory of self and others'. It aims to build a better understanding of what influences teaching in mathematics classrooms in order to inform teacher development. The study combines quantitative and qualitative methods through the use of questionnaires, interviews and a reflective research journal over a two year period and includes:  Secondary analysis of interviews with 20 teachers in schools with high numbers of students studying mathematics post-16;  Course evaluations from 27 teachers attending a workshop on CI and interviews with a sample who were willing to use the approach;  Pre and post study interviews with a lead mathematics teacher at two contrasting schools; one using CI with mixed ability groups and the other not.  Questionnaires completed by 221 Year 7 students and their mathematics teachers at the two contrasting schools. Open coding analysis of the teacher interviews was used to produce themes. The questionnaires were statistically analysed to explore teachers' and students' frameworks of intelligence and personality in relation to learning and performance goals in mathematics. The findings support the notion of a ‘dominant cultural script' for teaching mathematics in English secondary schools. Teachers refer to ‘expected national norms', where the expectations are driven by their understanding of National Strategy/Ofsted guidelines and the judgements upon them are based upon students' exam performance. This performance goal orientated model, coupled with teachers' anxieties about unacceptable behaviour in the classroom together with concerns about finding time to plan and resource a different approach, offers strong reasons for teachers' reluctance to change. The findings demonstrate that the teachers using CI still adhered, to some extent, to aspects of the ‘dominant cultural script'. They felt vulnerable in terms of examination results and inspection. The extent to which they deviated from the ‘script' was contingent upon factors such as having a strong supportive department with collaborative sharing of resources; seeing students as actively involved in the learning process and continuing professional development opportunities both within their schools and with university departments of education. Whilst these teachers, though mindful of exam performance and inspection, held other beliefs and goals for their students, these were not necessarily shared by the students. A high proportion of students, particularly amongst the lowest attaining students and girls, were found to hold fixed frameworks of intelligence and personality coupled with a preference for performance over challenge in mathematics. Dweck (2000) suggests that having such beliefs is unlikely to lead to mastery orientated qualities in students, which are the key to improvement in progress. Hence, given a dominant script for teaching mathematics which also emphasises performance goals, the likelihood of all students achieving their full potential in mathematics in such a climate is jeopardised.

Η χρήση αναφορών από την Ιστορία της Επιστήμης στα σχολικά εγχειρίδια γίνεται μέσα από μια ιδιότυπη ερμηνεία των ιστορικών πηγών από τους συγγραφείς τους, εντός των διαφορετικών κοινωνικών και πολιτισμικών πλαισίων κάθε εποχής, η οποία στοχεύει στην εξυπηρέτηση διδακτικών αναγκών και για το λόγο αυτό την ονομάζουμε διδακτική ανακατασκευή της Ιστορίας της Επιστήμης. Μία σημαντική περίπτωση διδακτικής ανακατασκευής που μελετάμε είναι η καθιέρωση της ονομασίας Θεώρημα του Θαλή στην ευρωπαϊκή και ειδικότερα στη νεοελληνική εκπαίδευση, της οποίας κύριο χαρακτηριστικό είναι ότι αντιστοιχεί σε διαφορετικά θεωρήματα στα εγχειρίδια διαφόρων ευρωπαϊκών χωρών. / Textbook writers uses their own interpetation for the references from the History of Science, through different social and cultural conditions of every era, an interpetation which serves purposes and objectives of the teaching of Science and for this reason we call it didactical reconstrution of the History of Science. An important case of didactical reconstruction which is the main theme our study is the establishment of the name Theorem of Thales in European and especially in Modern Greek education, a name that have as main feature that corresponds to differen ttheorems in the textbooks of different European countries.

Η ανθρώπινη σκέψη χαρακτηρίζεται από τη χρήση ειδών αναπαράστασεων για την ίδια έννοια και από τη δυνατότητα προσφυγής σε πολλαπλά συστήματα αναπαράστασης. Ως εκ τούτου, η ιδέα της αναπαράστασης αποτελεί βασικό εργαλείο της Σύγχρονης Διδακτικής των Μαθηματικών αφού οι αναπαραστάσεις θεωρούνται σύμφυτες με τα μαθηματικά. Οι αναπαραστάσεις που χρησιμοποιούνται στη μαθησιακή διαδικασία καθορίζουν σε σημαντικό βαθμό τα όσα μαθαίνει ο μαθητής και το πόσο εύκολα επιτυγχάνεται η κατανόηση των εννοιών στα μαθηματικά. Ο ρόλος αυτός αποκτά ιδιαίτερο ενδιαφέρον, όταν αναφερόμαστε στα σχολικά βιβλία αφού, αυτά εξακολουθούν να έχουν το κύριο βάρος στη μαθησιακή και διδακτική διαδικασία τόσο σε διάφορες χώρες, όσο και στην Ελλάδα ιδιαίτερα. Στην παρούσα εργασία καταγράφονται οι τύποι εξωτερικής αναπαράστασης και η συχνότητα εμφάνισής τους στα σύνθετα προβλήματα στα κεφάλαια που αναφέρονται στις πράξεις των ακεραίων αριθμών. Για το σκοπό αυτό μελετώνται τα σχολικά εγχειρίδια του παλιού και του νέου Αναλυτικού Προγράμματος Σπουδών. Η εργασία έχει δομηθεί σε τέσσερα μέρη: στο πρώτο μέρος γίνεται προσπάθεια να αποσαφηνιστούν οι όροι "αναπαράσταση" και "εξωτερικές αναπαραστάσεις" γενικά και στο χώρο των μαθηματικών ειδικότερα. Στο δεύτερο μέρος περιγράφεται ο ερευνητικός σχεδιασμός και ο τρόπος επεξεργασίας των εμπειρικών δεδομένων. Στο τρίτο μέρος παρουσιάζονται τα αποτελέσματα της επεξεργασίας των δεδομένων σε δύο άξονες (οριζόντιος και κάθετος. Η εργασία ολοκληρώνεται με το τέταρτο μέρος στο οποίο γίνεται προσπάθεια σύνοψης και γενικού σχολιασμού των αποτελεσμάτων και την εξαγωγή συμπερασμάτων / -

Τα τελευταία χρόνια πολλοί ερευνητές της διδακτικής των μαθηματικών έχουν εστιάσει στον ρόλο της πολλαπλής επίλυσης προβλήματος (επίλυση προβλήματος με περισσότερους από έναν τρόπους) στην ανάπτυξη της μάθησης του αντικειμένου. Δύο είδη πολλαπλής επίλυσης προβλήματος που μπορούν να ενταχθούν στη διδακτική πρακτική είναι καταρχήν αυτό με βάση το αποκαλούμενο “πρόβλημα πολλαπλών επιλύσεων” (“multiple solution task”) (Leikin et al., 2006) και δεύτερον αυτό με απαρχή το “πρόβλημα διασύνδεσης” (“interconnecting task”) (Kondratieva, 2011). Η διπλωματική αυτή εργασία θα περιέχει μια κριτική παρουσίαση των ποιοτικών χαρακτηριστικών των δύο προαναφερθέντων ειδών προβλημάτων, στο πλαίσιο των βασικών προσεγγίσεων επίλυσης προβλήματος που ανέπτυξαν οι Schroeder & Lester (1989) και Mamona-Downs & Papadopoulos (2006), καθώς και τα αποτελέσματα μιας πρώτης διερευνητικής προσπάθειας εφαρμογής αυτής της διδακτικής πρακτικής, σε τρεις τάξεις μαθηματικών σε σχολεία των Πατρών. Στην διπλωματική εργασία παρουσιάζονται ερευνητικά αποτελέσματα, αναφορικά με τις αιτίες της μη συστηματικής ένταξης της πολλαπλής επίλυσης προβλήματος στη διδακτική πρακτική. Επιπλέον, αναφέρονται συγκεκριμένες διδακτικές προσεγγίσεις για την πραγμάτωση της πολλαπλής επίλυσης προβλήματος στη διδασκαλία των μαθηματικών. Τέλος, σχολιάζονται κριτικά επιλεγμένες απαντήσεις συμμετεχόντων στην έρευνα, που αναδεικνύουν αφενός τα ποιοτικά χαρακτηριστικά της πολλαπλής επίλυσης προβλήματος, αφετέρου τον προσεκτικό σχεδιασμό που απαιτεί αυτή η πρωτοεμφανιζόμενη στην ελληνική βιβλιογραφία διδακτική πρακτική. / In recent years many researchers of mathematics’ education have focused on the role of multiple problem solving (problem solving in more than one ways) in the development of subject knowledge. Two types of multiple problem solving, that can be integrated into teaching practice, are firstly the "multiple solution task" (Leikin et al., 2006) and secondly the "interconnecting task" (Kondratieva, 2011). This thesis contains a critical presentation of the qualitative characteristics of these two types of problems, within the framework of the basic problem solving approaches developed by Schroeder & Lester (1989) and Mamona-Downs & Papadopoulos (2006), and the results of a first exploratory effort to implement this teaching practice, in three mathematics’ classrooms in schools of Patras. In this thesis we present research results regarding the causes of non-systematic implementation of multiple problem solving in teaching practice. Moreover, we outline specific teaching approaches for the implementation of multiple problem solving in mathematics’ teaching. Finally, we critically comment selected responses of participants in the survey, that highlight both the qualitative characteristics of multiple problem solving and the careful planning that is required for this nascent teaching practice in the Greek literature.

The thesis presents the analytical process and the findings of a study on: lecturers teaching practice with first year undergraduate mathematics modules; and lecturers knowledge for teaching with regard to students mathematical meaning making (understanding). Over three academic semesters, I observed and audio-recorded twenty-six lecturers teaching to a small group tutorial of two to eight first year students, and I discussed with the lecturers about their underlying considerations for teaching. The analysis of this thesis focuses on a characterisation of each of three (of the twenty-six) lecturers teaching, which I observed for more than one semester. I chose the teaching of three experienced lecturers, due to diversity in terms of ways of engaging the students with the mathematics, and due to my consideration of their commitment to teaching for students mathematical meaning making. The distinctive nature of the study is concerned with the conceptualisation of university mathematics teaching practice and knowledge within a Vygotskian perspective. In particular, I used for the characterisation of teaching practice and of teaching knowledge the notions tool-mediation and dialectic from Vygotskian theory. I also used a coding process grounded to the data and informed by existing research literature in mathematics education. I conceptualised teaching practice into tools for teaching and actions with tools for teaching (namely strategies). I then conceptualised teaching knowledge as the lecturers reflection on teaching practice. The thesis contributes to the research literature in mathematics education with an analytical framework of teaching knowledge which is revealed in practice, the Teaching Knowledge-in-Practice (TKiP). TKiP analyses specific kinds of lecturer s knowing for teaching: didactical knowing and pedagogical knowing. The framework includes emerging tools for teaching (e.g. graphical representation, rhetorical question, students faces) and emerging strategies for teaching (e.g. creating students positive feelings, explaining), which were common or different among the three lecturers teaching practice. Overall, TKiP is produced to offer a dynamic framework for researcher analysis of university mathematics teaching knowledge. Analysis of teaching knowledge is important for gaining insights into why teaching practice happens in certain ways. The findings of the thesis also suggest teaching strategies for the improvement of students mathematical meaning making in tutorials.

The research reported in this thesis investigated staff-student collaboration in advanced undergraduate mathematics course design and delivery at a research-intensive UK university. Staff and students collaborated to redesign and deliver two courses: Vector Spaces and Complex Variables. The collaboration in the design of the two courses involved students who had completed the courses and then who worked as interns together with a small team of academic staff. The collaboration in the delivery of the two courses involved the implementation of a Peer Assisted Learning (PAL) scheme in which third-year students facilitated the learning of second-year students in optional scheduled sessions. The study employed a mixed-methods research strategy involving an ethnographic approach to the study of the course design process and PAL sessions followed by an observational study (a quasi-experimental design) to investigate the impact of PAL attendance on the achievement of PAL participants. This thesis reports findings from a three-phase research design. Phase one explored the nature of the collaborations in course design and its impact on staff teaching practices and on the student collaborators. Phase two investigated the characteristics of the PAL sessions for the advanced undergraduate mathematics courses and the roles played in those sessions. Phase two also explored the impact of PAL in qualitative terms on both PAL participants and PAL leaders. Phase three investigated the impact of PAL in quantitative terms on the achievement of students who participated as PAL participants. The study found that staff-student collaboration in course design and delivery led to emergent Communities of Practice in which staff and students engaged in mathematics practice which led to identity transformation of student collaborators, a deeper understanding of the mathematics on which the students worked and some change in staff teaching and course design practice. The also showed that staff-student collaboration in the delivery of course units via PAL resulted in a learning community in which PAL participants and PAL leaders engaged in mathematics practice which led to increased student achievement and enhanced affective outcomes for both PAL participants and PAL leaders.

Adolescent students’ perceptions of parental influence in relation to mathematics education is an under-researched area, since most studies in this area focus on parental involvement in primary mathematics. This research study aims to fill this gap in the literature by exploring adolescent students’ perceptions of parental influence on their dispositions towards studying mathematically-demanding courses in Higher Education (HE). This study employs mixed research methods to investigate students’ perceptions of parental influence with a survey (N=563) and semi-structured interviews with six students and their parents. Additional interviews were conducted with three immigrant families in Cyprus. The study builds on Bourdieu’s theory of reproduction of social inequalities and extends some concepts of Bourdieu’s theoretical framework to discuss parental influence. The main findings of this research study are reported in three journal papers whilst the papers are under review for publication. The first paper of the thesis reports the analysis of the quantitative data of this study by combining Rasch analysis with statistical modelling. The statistical analysis showed that parental influence is not statistically significant for predicting students’ mathematical dispositions in some models, when other background variables are included in the models. However, further statistical analysis showed that the effect of parental influence is mediated through students’ choice of mathematics course and their mathematical inclination. The non-statistical significance of parental influence in some models was interpreted as a ‘misrecognition’ after Bourdieu (1980). The same phenomenon was noted in the second paper of the thesis, where students and their parents ‘denied’ parental influence during their interviews. This was again interpreted as ‘misrecognition’ and parental influence is conceptualised as a form of ‘symbolic violence’ that parents exercise on their children. Arguably, parents possess more capital in the family field and their influence on their children’s educational choices might be unconscious, thus students’ misrecognise their parents’ influence but they draw significantly on their family’s capital to make their choices for future studies in HE.Lastly, the third paper of the thesis explored cultural differences in students’ perceptions of parental influence in England and Cyprus; stronger perceptions of parental influence were identified in immigrant students’ interviews compared to indigenous students in both countries. Bourdieu’s (1977) concept of ‘hysteresis’ was adapted to theorise this phenomenon. Arguably, while immigrant students’ habitus adjust to the new field, students become more reflective on their parents’ influence because of the reflexivity the hysteresis effect entails. In all three papers Bourdieu’s theoretical framework was used to operationalise students’ mathematical dispositions and to interpret the findings of this study. The main contributions to knowledge of this study is the operationalisation of students’ mathematical habitus; the new theoretical conceptualisation of parental influence as a form of symbolic violence in the family field and the extension of the hysteresis effect to interpret immigrant students’ stronger perceptions of their parents’ influence.